Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/376

Let $P_{1},P_{2},\ldots P_{n}$ be the potentials at $A_{1},A_{2},\ldots A_{n}$ respectively, and let $Q_{1},Q_{2},\ldots Q_{n}$ be the quantities of electricity which enter the system in unit of time at each of these points respectively. These are necessarily subject to the condition of 'continuity'

$Q_{1}+Q_{2}\ldots +Q_{n}=0,$

(4)

since electricity can neither be indefinitely accumulated nor produced within the system.

The condition of 'continuity' at any point $A_{p}$ is

$Q_{p}=C_{p1}+C_{p2}+\mathrm {\&c.} +C_{pn},$

(5)

Substituting the values of the currents in terms of equation (1), this becomes

${\begin{aligned}Q_{p}=(K_{p1}+K_{p2}+\mathrm {\&c.} +K_{pn})P_{p}-(K_{p1}P_{1}+K_{p2}P_{2}+\mathrm {\&c.} +K_{pn}P_{n})\\+(K_{pq}E_{p1}+\mathrm {\&c.} +K_{pn}E_{pn}).\end{aligned}}$

(6)

The symbol $K_{pp}$ does not occur in this equation. Let us therefore give it the value

$K_{pp}=-(K_{p1}+K_{p2}+\mathrm {\&c.} +K_{pn});$

(7)

that is, let $K_{pp}$ be a quantity equal and opposite to the sum of all the conductivities of the conductors which meet in $A_{p}$. We may then write the condition of continuity for the point $A_{p}$,

${\begin{aligned}K_{p1}P_{1}+K_{p2}+\mathrm {\&c.} +K_{pn}P_{n}\qquad \qquad \\=K_{p1}E_{p1}+\mathrm {\&c.} +K_{pn}E_{pn}-Q_{p}.\end{aligned}}$

(8)

By substituting 1, 2, &c. n for p in this equation we shall obtain n equations of the same kind from which to determine the n potentials $P_{1},P_{2},{\&c.},P_{n}.$

Since, however, there is a necessary condition, (4), connecting the values of $Q,$ there will be only $n-1$ independent equations. These will be sufficient to determine the differences of the potentials of the points, but not to determine the absolute potential of any. This, however, is not required to calculate the currents in the system.

If we denote by $D$ the determinant

$D={\begin{array}{|lll|}K_{11},&K_{12},&\ldots \ldots K_{1(n-1)},\\K_{21},&K_{22},&\ldots \ldots K_{2(n-1)},\\----&----&--------\\K_{(n-1)1},&K_{(n-1)2},&\ldots \ldots K_{(n-1)(n-1)};\end{array}}$

(9)

and by $D_{pq},$ the minor of $K_{pq},$ we find for the value of $P_{p}-P_{n},$

${\begin{aligned}(P_{p}-P_{n})D=(K_{12}E_{12}+\mathrm {\&c.} -Q_{1})D_{p1}+(K_{21}E_{21}+\mathrm {\&c.} -Q_{2})D_{p2}+\mathrm {\&c.} \\+(K_{q1}E_{q1}+\mathrm {\&c.} +K_{qn}E_{qn}-Q_{q})D_{pq}+\mathrm {\&c.} \qquad \end{aligned}}$

(10)

In the same way the excess of the potential of any other point,